"Degeneracy" is one of those words that mean something quite different when used by a preacher, a chess player, an astrophysicist, or a mathematician.* To chemists and physicists, degeneracy describes a state in which an electron could potentially occupy either of two orbital paths around a molecule, both of which have the same energy level.

Under a scanning tunneling microscope, carbon-60 buckyballs in a single layer appear triangular when doped with three potassium atoms (K3C60), which renders the layer metallic, but appear nearly rectangular when doped with four potassium atoms (K4C60), which makes the carbon layer insulating.

Nature likes to avoid such ambiguities, especially when it can lower the overall energy of the system, and the Jahn-Teller effect (or Jahn-Teller distortion) is one means to that end. The Jahn-Teller effect occurs when a new electron is placed in a molecule and lands in one of these equal-energy orbitals. Such an electron tends to distribute itself around the atomic nuclei in a way that causes some atoms to move closer together and others farther apart. One result of this structural deformation is that the equal energy levels of the original orbitals are split apart: the one occupied by the new electron now has slightly less energy, and this lowers the energy of the molecule as a whole.

"The Jahn-Teller effect is important because it addresses what structural changes occur when charge moves into a molecule," says Michael Crommie of Berkeley Lab's Materials Sciences Division, who is a professor of physics at UC Berkeley. "It affects even high-temperature superconductivity, for example, and is critical for determining spin-orbit coupling in magnetic molecules. It's fundamental to understanding the charge-structure relationship at nanometer length scales," where distances are measured in billionths of a meter.

But while collective Jahn-Teller effects have been observed by ensemble experimental techniques like optical spectroscopy, which average over many molecules, these effects have never been observed at the single-molecule level until now.

In experiments with a single layer of carbon-60 molecules, buckyballs, coating a substrate of gold and doped with varying proportions of potassium atoms, Crommie and his colleagues used a scanning tunneling microscope (STM) at very low temperature to obtain images of what happens as electrons are added to each buckyball. On average, each potassium atom donates a single electron to a neighboring buckyball.

"Because we are looking at a two-dimensional layer of C60, we can see things that would be impossible to see in bulk materials," Crommie says. "We can directly observe how single-molecule behavior drives the collective behavior of the material."

In STM microscopy, variations of current between the microscope's miniscule tip and the surface of the sample can be used to provide both a topographical image  each atom or molecule shows up as a characteristically shaped bump  and also to measure its electronic structure, by measuring the local density of occupied or unoccupied electronic states at different voltages  the greater the density of electronic states, the brighter the image at that microscopic location.

For this experiment the researchers evaporated potassium onto the buckyball monolayer for a length of time previously determined to produce the desired proportion of dopant atoms. They arranged to have part of the sample doped with three potassium atoms per buckyball, forming the compound K3C60, and another part doped with four potassium atoms per buckyball, K4C60.

In the field of the scanning tunneling microscope, patches of both K3C60 and K4C60 appear side by side, exhibiting different, distinctively ordered patterns.

"We saw uniform behavior for average dopant levels of K3C60 or K4C60, and when the average number of potassium atoms per C60 was somewhere between three and four, then the pattern divided into regions having one dopant level or the other. So we could be pretty sure what we were looking at," Crommie says.

Pure C60 is an insulator, with its highest occupied molecular orbitals (which resemble a bulk semiconductor's valence bands) full of electrons, 10 in all, and its lowest unoccupied orbitals (resembling conducting bands) empty. Adding three electrons half-fills the unoccupied bands and turns the molecule into a metallic conductor.

But when you add one more electron's charge per molecule (by increasing the potassium atoms per buckyball from three to four) the state of the molecular layer flips from a metallic conductor back to an insulator, with vivid changes in molecular structure clearly visible in the STM images.

The Crommie group's images of metallic K3C60 showed a lattice of triangles whose appearance did not vary much when the voltage was adjusted to bring out either the highest occupied orbitals or the lowest empty ones. This is typical of a metal, whose conducting electrons are in far-ranging, continuous motion.

The triangular appearance of some of the buckyballs, Crommie says, is due to the fact that the greatest density of states occurs in close-spaced rings surrounding the buckyball's 12 pentagons (a buckyball's 60 carbon atoms mark the vertices of 12 pentagons and 20 hexagons, in a shape like a soccer ball).

"These rings can be used to determine the orientation of the buckyball in the STM image, information that is essential to theorists for interpreting the molecules' behavior," he says. From any given point of view, three of the rings are likely to be prominent, which accounts for the triangle shapes, all of which are regularly oriented.

The difference in the appearance of insulating K4C60 was startling: now the individual molecules arranged themselves in groups of four, oriented toward one another at precise angles. Even more striking was the difference between molecular images of filled and empty electronic states, the former bisected by a single dark line and the latter by two such lines at right angles. These were clear evidence of the splitting of the energy levels of the two types of orbitals  the hallmark of the Jahn-Teller effect.

When four electrons are added to a buckyball (as from doping with potassium atoms), the Jahn-Teller distortion causes flattening, splitting degenerate energy levels into higher energy empty states and lower energy filled states, visibly distinguished by the scanning tunneling microscope.

In contrast to the case of K3C60, where the three extra electrons occupy the molecule's otherwise empty "conduction" band, adding a fourth electron causes the occupied and empty bands to split, and all four electrons completely fill the lower energy band.

Crommie and his Berkeley colleagues, including theorist Steven Louie, calculated from first principles the structural distortion that a C60 molecule with four extra electrons could undergo. Four stable structures were possible, but only one of these could produce the electronic wave functions revealed in the STM images.

In this way the researchers were able to understand how changing the buckyball's charge from three to four electrons induces the Jahn-Teller effect to flatten the near-spherical buckyball, resulting in a splitting of the energy levels of filled and empty orbitals and changing the two-dimensional C60 monolayer from a conducting metal to an insulator.

* Degeneracy as "moral perversion" is the familiar meaning. Less familiar is the word's meaning in chess, where it refers to a game that cannot be won by either player with the remaining pieces; in astrophysics, where it refers to matter so dense that atoms have collapsed; or in mathematics, where it refers to a limit of a set which is a different, usually simpler set: for example, a point is a circle with no radius. And there are other definitions. See Wikipedia.